I think it really depends on the nickname. Personally I find
Abby (with any spelling) to be too abbreviated. I also find Maddy to be this way too.
Alex is ok for a boy, but I think that the name
Alex is still a bit to masculine for it to just be a girl's full name.
Katy and
Lily are alright. Overall I think that the point of a nickname is to give a child an alternative to a fuller, more formal name that they may choose to use when they are older. Also, remember that for the rest of her life people will try to call her the full form of her name and she will constantly have to be correcting them, or saying "No, it's just
Abbey".